Electrodynamic Levitation (EDL), operating on the principles of electromagnetic induction and repulsive Lorentz forces, presents a cornerstone technology for enabling contactless, high-speed transportation. This review systematically synthesizes the scientific literature concerning the application of EDL systems, with a particular focus on the dynamic stability challenges inherent to their operation at very high speeds. The passive nature of EDL, while advantageous for its simplicity and reliability, introduces unique oscillatory dynamics in both vertical and lateral directions, coupled with parasitic effects like eddy current drag and Joule heating. This paper meticulously classifies these stability issues, which include underdamped oscillations, hunting instability, and complex vehicle-guideway interactions.